Neutron Scattering from Magnetic Materials part 9 docx

Spherical neutron polarimetry 237 Fig. 10. The h0l section of reciprocal space for the incommensurate structure of CuO. The fundamental (nuclear) reflections are shown as open circles and the magnetic satellites as filled ones. The scattering vectors for the 002 +τ and 000 −τ reflections are shown as dashed lines. 4.4. Magnetic structures with zero propagation vector It has already been pointed out in Section 2.3 that in structures with τ = 0, nuclear and magnetic scattering occur in the same Bragg reflections and interference between them can occur. In this case SNP can determine the ratio between magnetic and nuclear scattering, allowing the magnitude of magnetic moments to be established from SNP alone without recourse to supplementary integrated intensity measurements. 4.4.1. The magnetic structure of U 14 Au 51 . The intermetallic compound U 14 Au 51 crys- tallises in the hexagonal Gd 14 Ag 51 structure with space group P 6/m [11]. The uranium atoms occupy three crystallographically distinct sites 6(k), 6(j ) and 2(e) labelled U1, U2 and U3, respectively. Susceptibility, specific heat and resistivity indicate a magnetic phase transition at 22 K. This has been confirmed by neutron powder diffraction measure- ments: an antiferromagnetic structure with zero propagation vector and magnetic space group P 6  /m was proposed [12]. The ordered magnetic moments aligned parallel to c were 0.5 and 1.6µ B on U1 and U2, respectively. No moment was assigned to the U3 atoms which have a particularly small separation, it was argued that direct f -electron wave func- tion overlap prevents a magnetic response. An SNP study of the magnetic structure was undertaken because it proved impossible to reconcile the intensity of magnetic scattering by single crystals of U 14 Au 51 with the proposed magnetic structure. The U 14 Au 51 crys- tal was mounted with its [01.0] axis vertical. The polarisation matrices determined for the 20.0, 20.1 and 10.1 reflections are given in Table 1. They enable severe constraints to be imposed on the possible magnetic structures. 1. For incident polarisation parallel to the scattering vector x the scattered beam is partly depolarised, and reversed but not rotated. The depolarisation must be due to off- diagonal terms (P xy , P xz ) of opposite signs coming from 180 ◦ domains. This means that the J ni terms in (14) must be nonzero showing that the magnetic scattering is 238 P. J . B r o w n Table 1 Polarisation matrices P ij measured for the 20.0, 20.1 and 10.1 reflections of U 14 Au 51 at 15 K hkl P ij =20.0 P ij =20.1 P ij =10.1 j i xyzxyzxyz x −0.86 0 0 −0.28 0 0 −0.54 −0.08 −0.07 y 0 −0.83 0 0 −0.62 −0.38 −0.08 0.98 −0.12 z 0 0 0.98 0 0.37 0.73 0.01 −0.05 −0.55 in quadrature with the nuclear scattering. The reversal of direction shows that the magnetic structure factor is greater than the nuclear one for both 20.0 and 20.1. 2. For the 20.0 reflection there are no significant off-diagonal terms and P zz is not sig- nificantly different from 1 showing that for 20.0 M ⊥ must be parallel to z (crystal- lographic [01.0]), and there are therefore no significant components of the magnetic structure factor parallel to c. 3. For 20.1 there is some depolarisation for all three incident polarisation directions and off-diagonal components P yz and P zy are observed. This is consistent with moments in the a–b plane. The observation that the depolarisation for incident directions in the y–z plane, that containing the magnetic interaction vector, is less than for the x direction implies that all the depolarisation is due to the 180 ◦ domains and that there is none due to orientation domains. The magnetic structure therefore probably has the full symmetry of the crystallographic space group. There is just a single magnetic space group and basic model structure which is compati- ble with all these constraints. The U1 and U2 sites lie on the mirror planes perpendicular to the hexad and from (2) and (3) their moments lie in it. These mirror planes cannot therefore invert the moments and must be combined with time reversal. To satisfy (1) which implies that centrosymmetrically related atoms have opposite moments the hexad must operate without time inversion. The magnetic space group is therefore P 6/m  and the magnetic moments on the groups of 6 U1 (and U2) atoms related by the hexad have a star struc- ture as illustrated in Figure 11. In this magnetic group any moment on the U3 atoms is constrained to be parallel to c since these sites are on the hexagonal axes. The SNP mea- surements show that the c component of moment is small or zero so it can be concluded that the U3 moment is also small or zero. To describe the structure completely it is neces- sary to determine the magnitude and the orientation within the a–b plane of the moments on the U1 and U2 atoms. The SNP data for the 10.1 reflection (Table 1) allow rough values of the moment direc- tions within the a–b plane to be deduced. The incident polarisation parallel to y is hardly changed on scattering so its magnetic interaction vector M ⊥ is nearly parallel to y.The magnitude of M ⊥ can be obtained from P xx = 1 −γ 2 1 +γ 2 =β, (27) γ gives, as before (equation (10)), the ratio of magnetic to nuclear scattering. M ⊥ for Spherical neutron polarimetry 239 Fig. 11. The hexagonal star arrangement of moments found in the U1 and U2 layers of U 14 Au 51 . The angle φ used to fix the orientation of the moments is marked. the 10.1 reflection is the sum of contributions from the U1 and U2 layers. The y and z components of these were computed separately as a function of the angle φ in Figure 11 and are plotted in Figure 12. By moving one curve relative to the other it was found that there is only a small range in which a pair of φ’s exist for which the M ⊥z for U1 and U2 cancel whilst their M ⊥y reinforce one another. It corresponds to φ U1 ∼ 140 ◦ , φ U2 ∼ 90 ◦ . These initial values provided an adequate starting point for a least squares refinement of the structure using both SNP and integrated intensity data. 4.4.2. Magnetoelectric crystals. The property of magnetoelectricity in centrosymmetric crystals is restricted to those having antiferromagnetic structures with zero propagation vector in which the centre of symmetry is combined with time-reversal. These are just the requirements for J ni (equation (16)) to be finite giving rise to off-diagonal terms P xz , P zx in the polarisation matrix (equation (14)). It is known that although the temperature dependencies of magnetoelectric (ME) susceptibilities are unique to each material, their magnitudes and even their signs are specimen dependent. This specimen dependence is due to the existence of 180 ◦ antiferromagnetic domains which have opposite ME effects. The measured ME susceptibility χ obs is related to the intrinsic susceptibility χ 0 by χ obs =ηχ 0 with η = v 1 −v 2 v 1 +v 2 , where v 1 and v 2 are the volumes of crystal belonging to each of the two domains. SNP gives the possibility, for the first time, to obtain the intrinsic ME susceptibilities since it allows the domain fraction η to be determined. For a centrosymmetric ME crystal with domain fraction η and moments in the x–y plane the polarisation matrix (equations (14) and (10)) can be simplified to P =  β 0 ηξ 010 −ηξ 0 β  β is given by equation (27) and ξ = 2q y γ/(1 + γ 2 ), (28) 240 P. J . B r o w n Fig. 12. Curves showing the variation with φ of M ⊥y (full) and M ⊥z (dashed) components of the magnetic interaction vector of the 10.1 reflection of U 14 Au 51 . (a) is for the U1 atoms and (b) for the U2 atoms. The origins of the two figures are displaced so that on the vertical line marked ((a) φ =140 ◦ ,(b)φ = 90 ◦ )thez components of U1 and U2 cancel whilst their y components reinforce one another. Fig. 13. The moment directions of Cr 3+ ions at the centres of the double octahedral coordination polyhedra found in Cr 2 O 3 , after (a) cooling in parallel, (b) antiparallel electric and magnetic fields. q y is +1ifM(Q) is parallel to y and −1 if it is antiparallel. Measurement of the polar- isation matrix therefore allows both η and γ to be determined. The absolute directions of rotation of the neutron spins when η = 0 determine the magnetic configuration of the more populous domain. This in turn allows the effects of electric and magnetic fields on the domain population to be studied. The results shed light on the fundamental mecha- nisms leading to the ME effect. Perhaps the best known ME material is Cr 2 O 3 in which the Cr 3+ ions are octahedrally coordinated by oxygen and the structure is made up of pairs of octahedra, sharing a common face as illustrated in Figure 13, linked to other pairs by sharing the free vertices. SNP has shown that electric and magnetic fields, applied parallel to one another and to the c axis while cooling through the Néel transition, stabilise the Spherical neutron polarimetry 241 domain in which the moments point towards the shared face of their coordinating oxygen octahedra [13]. 5. Determination of antiferromagnetic form factors As has been shown in Chapter 4 the classical polarised neutron diffraction technique [14] is widely used to study the magnetisation distribution around magnetic atoms and ions in ferromagnetic and paramagnetic materials. It is very much more difficult to measure this distribution in antiferromagnetic systems because in antiferromagnets the cross-section is seldom polarisation dependent so the classical method is not applicable. As a consequence, very few measurements of magnetisation distributions in antiferromagnetic materials have been made since usually they require very precise integrated intensity measurements of rather weak reflections. In the few cases where such measurements have been undertaken, they have given very interesting results [15–17]. An antiferromagnetic magnetisation dis- tribution is more sensitive than a ferromagnetic or paramagnetic one to the effects of cova- lency because the overlap of positive and negative transferred spin on the ligand ions leads to an actual loss of moment rather than just to a redistribution. Until recently no precise measurements had been made for the class of antiferromagnetic structures with zero propagation vector, in which magnetic atoms of opposite spin are re- lated by a centre of symmetry. In such structures the magnetic and nuclear scattering are superimposed, making separation of the nuclear and magnetic parts difficult. Additionally the magnetic and nuclear structure factors are in phase quadrature so that is no interfer- ence between them to give a polarisation dependent cross-section. However, it was shown in the previous section that it is in exactly this case for which the magnetic and nuclear scattering are in quadrature that the polarisation matrices depend sensitively on the ratio γ between the magnetic and nuclear structure factors when there is an imbalance (η = 0) in the populations of the two 180 ◦ domains. The high precision with which the ratio γ can be determined in favourable cases allows the magnetic structure factors to be determined with good accuracy and so gives access to the antiferromagnetic form-factors. The polarisation matrix of (28) allows two independent estimates of γ : (a) P xz =−P zx =ηξ = ηq y γ 1 +γ 2 , (29) (b) P xx =P zz =β = 1 −γ 2 1 +γ 2 , the former only being useful if there is an imbalance in the 180 ◦ domains. Assuming the polarimeter (CRYOPAD) is free of aberrations the precision with which γ can be deter- mined depends on the statistical error in the determination of the components of scattered polarisation. It should be recalled that in this type of structure the cross-section is inde- pendent of the polarisation direction. The counting rate summed over the two polarisation states accepted by the detector is therefore constant, and independent of either incident or scattered polarisation direction. The polarisation measured by the analyser is given by P = (I + − I − )/(I + + I − ) where I + and I − are the counting rates in the two detector 242 P. J . B r o w n channels. The variance in the measurement of a component of polarisation due to counting statistics is V P = (1 −P 2 ) 2 4  1 N + + 1 N −  , (30) where N + and N − are the counts recorded in each channel. The variance is minimised by dividing the measuring time available in the ratio t + /t − =(1 −P)/(1 + P). With this division, if the total number of neutrons counted is N equation (30) becomes V P = 1 −P 2 N . (31) The variances in the values of γ derived from (29) are V γ = (1 +γ 2 ) 4 16γ 2 V P from (a) and (32) V γ = (1 +γ 2 ) 4 4η 2 (1 −γ 2 ) 2 V P from (b). If η is small (nearly equal domains) or γ is close to unity, the best estimate of γ will be obtained from (29)(a) whereas for very small or very large γ (29)(b) will give a better value so long as η is nonzero. Figure 14 shows the regions of γ –η space in which one or the other equation gives the better estimate of γ . The first example of the use of this technique was to determine the Cr 2+ form-factor in Cr 2 O 3 [18]. Samples were cooled in combined electric and magnetic fields to obtain dif- ferent domain ratios η as indicated in the previous section. The crystals were aligned with Fig. 14. Plot of γ –η space. The shaded region is that in which equation (29)(a) gives a more precise estimation of γ than (29)(b). The γ axis is plotted on a logarithmic scale. Spherical neutron polarimetry 243 Fig. 15. The experimental values of the magnetic form factor measured at the h0. Bragg reflections of Cr 2 O 3 . The smooth curve is the spin-only free-ion form factor for Cr 2+ normalised to the experimental value at the lowest angle reflection (10.2). a [ ¯ 12 ¯ 10] axis vertical so as to obtain reflections h0 ¯ hl in the horizontal plane. The moment direction is [0001] so that with this orientation M ⊥ (Q) is parallel to polarisation y.The elements P xz , P zx of the polarisation matrices obtained with different domain ratios are very different, but the magnetic structure factors deduced from them were found to agree well. This confirms the supposition that extinction effects are not a major problem since the measurements of polarisation are made with a constant cross-section. The points on the Cr 3+ form factor obtained from the measured structure factors are plotted in Figure 15 where they are compared with the Cr 3+ free in form factor. It can be seen that for most reflections an extremely good precision was obtained. Exceptions are the 20. ¯ 2 and 10. ¯ 10 reflections; for the former the nuclear structure factor is small so that γ  1 and in the latter the geometric structure factor for the Cr atoms is small so the reflection is insensitive to the Cr form factor. This first pioneering experiment has shown that SNP can be used to make high preci- sion measurements of antiferromagnetic magnetisation distributions. However is should be emphasised that such measurements are only possible for a restricted class of antiferro- magnets, those in which magnetic and nuclear scattering occur in quadrature in the same reflections. Additionally high sensitivity can only be obtained if the population of the 180 ◦ domains can be unbalanced. Nevertheless, this class of antiferromagnets includes the mag- netoelectric classes, and for these electromagnetic annealing can unbalance the domains. SNP therefore provides an important new tool for probing the magnetisation distributions associated with magnetoelectricity. 244 P. J . B r o w n References [1] R.M. Moon, T. Riste and W.C. Koehler, Phys. Rev. 181 920 (1969). [2] O. Schärpf, Physica B 182 376 (1992). [3] F. Tasset, P.J. Brown, E. Lelièvre-Berna, T. Roberts, S. Pujol, J. Alibon and E. Bourgeat-Lami, Physica B 267–268 69 (1999). [4] M. Blume, Phys. Rev. 130 1670 (1963). [5] S.V. Maleev, V.G. Bar’yaktar and P.A. Suris, Sov. Phys. Solid State 4 2533 (1963). [6] P.J. Brown, Physica B 297 198 (2001). [7] J.B. Forsyth, P.J. Brown and B.M. Wanklyn, J. Phys. C 21 2917 (1988). [8] D. Yablonski, Physica C 171 454 (1990). [9] Yu.G. Raydugin, V.E. Naish and E.A. Turov, J. Magn. Magn. Mater. 102 331 (1991). [10] P.J. Brown, T. Chattopadhyay, J.B. Forsyth, V. Nunez and F. Tasset, J. Phys.: Condens. Matter 3 4281 (1991). [11] A. Dommann and F. Hullinger, J. Less-Common Met. 141 261 (1988). [12] A. Dommann et al., J. Less-Common Met. 160 171 (1990). [13] P.J. Brown, J.B. Forsyth and F. Tasset, J. Phys.: Condens. Matter 10 663 (1998). [14] R. Nathans, C.G. Shull, G. Shirane and A. Andresen, J. Phys. Chem. Solids 10 138 (1959). [15] H.A. Alperin, Phys. Rev. Lett. 6 520 (1961). [16] J.W. Lynn, G. Shirane and M. Blume, Phys. Rev. Lett. 37 154 (1976). [17] X.L. Wang, C. Stassis, D.C. Johnstone, T.C. Leung, J. Ye, B.N. Harmon, G.H. Lander, A.J. Shultz, C K. Loong and J.M. Honig, J. Appl. Phys. 69 4860 (1991). [18] P.J. Brown, J.B. Forsyth, E. Lelièvre-Berna and F. Tasset, J. Phys.: Condens. Matter 14 1957 (2002). CHAPTER 6 Magnetic Excitations Tapan Chatterji Institut Laue–Langevin, B.P. 156X, 38042 Grenoble cedex, France E-mail: chatt@ill.fr Contents 1. Introduction . . . . . . . . . . . 247 2. Experimental methods . . . . . 247 2.1.Triple-axisspectrometer 247 2.2.IntensityandresolutionfunctionofTAS 251 2.3.Sizeandshapeoftheresolutionfunction 254 2.4. TAS multiplexing . . . . . 256 2.5.Time-of-flightspectrometers 256 3.Spinwavesinlocalizedelectronsystems 259 3.1.SpinwavesinHeisenbergferromagnets 259 3.2.ThermalevolutionofspinwavesinHeisenbergferromagnets 263 3.3.SpinwavedampinginHeisenbergferromagnets 265 3.4.SpinwavesinHeisenbergantiferromagnets 268 3.5. Two-magnon interaction in Heisenberg antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . 274 3.6.SpinwavesinHeisenbergferrimagnets 275 4. Spin waves in itinerant magnetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 4.1. Generalized susceptibility and neutron scattering cross-section . . . . . . . . . . . . . . . . . . . . 281 4.2. Spin dynamics of ferromagnetic Fe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 4.3. Spin dynamics of ferromagnetic Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4.4. Spin dynamics of weak itinerant ferromagnet MnSi . . . . . . . . . . . . . . . . . . . . . . . . . . 297 5.SpinwavesinCMRmanganites 300 5.1. Spin waves A 1−x B x MnO 3 ,A= La, Pr, Nd; B =Ca,Sr,Ba 300 5.2. Thermal evolution of spin dynamics of A 1−x B x MnO 3 313 5.3. Spin waves in bilayer manganite La 2−2x Sr 1+2x Mn 2 O 7 315 6.Concludingremarks 327 Acknowledgments 328 References 328 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji © 2006 Elsevier B.V. All rights reserved 245 [...]... next section Apart from the neutron source which is usually a reactor, the monochromator crystal (or assembly of crystals) is the most important component of the triple-axis spectrometer that determines the neutron intensity incident on the sample The monochromator crystal selects a specific neutron wavelength from the incident polychromatic neutron beam from the reactor by Bragg diffraction from a given... multiplied by (1 − Q2 ) is the longitudinal part of the cross-section, while the term multiplied z Magnetic excitations 261 by (1 + Q2 ) is the transverse part In the linear spin wave approximation S z (t) = S z (0), z therefore the longitudinal part leads to elastic scattering and does not concern us here So the transverse neutron scattering cross-section from spin waves from a crystal lattice at low temperature... experimentally determined by inelastic neutron scattering yielding eventually the sign and magnitudes of exchange interactions Before we explain how this may be achieved in particular cases we describe the essential experimental techniques of inelastic neutron scattering 2 Experimental methods As in the case of the determination of the magnetic structure, the neutron scattering technique is unique for... respectively The neutron absorption cross-sections of these isotopes are 92 09 ± 100 barn and 312 ± 7 barn, respectively The very high absorption coefficient of 151 Eu makes inelastic neutron scattering investigation impossible on EuS synthesized from natural Eu which has an absorption coefficient of 4565 ± 100 barn Therefore to reduce the absorption of neutrons, EuS single crystals were grown from melt of... to the scattered neutron energies to be analyzed and the energy resolution required The detector is generally a simple 3 He-gas proportional counter A counting efficiency of about 80 95 % in the relevant neutron energy range is achieved by choosing the thickness of the counter and the gas pressure Unlike the highly collimated X-ray beam from the synchrotron sources the neutron beam from the reactor emerge... time that a neutron takes to reach the sample from a known starting point and also that to cover the distance from the sample to the detector after the scattering process The total time is compared with the known flight time of the neutrons that are scattered from the sample elastically, i.e., without change in energy For this purpose one needs a sharp neutron pulse with known start time and location... spectrometer a polychromatic beam hits the sample, but only a fixed neutron velocity (energy) is accepted after the scattering process In this case the flight time from the sample to the detector is known So from the total flight time from the chopper to the detector it is possible to determine the initial energy and the energy transfer caused by the scattering process TOF instruments for which the monochromatic... y y The neutron scattering cross-section can be rewritten in terms of spin raising and lowering operators as d 2σ = dΩ dE γ e2 mc2 × 2 1 gF (Q) 2 1 − Q2 z + 1 + Q2 z × 1 2πh ¯ 1 2πh ¯ 2 k k ∞ −∞ ∞ −∞ exp −2W (Q) z z dt exp(−iωt) SQ (0)S−Q (t) (26) dt exp(−iωt) 1 + − − + S (0)S−Q (t) + S−Q (0)SQ (t) 4 Q (27) Here the neutron scattering cross-section has been broken into two parts: (1) the part multiplied... unique for experimental investigation of spin waves and other excitations in magnetic crystals The spin wave energies in magnetic solids are normally in the meV range and therefore scattering of thermal neutrons is suitable for their investigation Sometimes the spin wave energy is of the order of 0.1 meV for which cold neutron scattering is more appropriate In some cases like transition metal ferromagnets... structure with lattice parameter a = 5 .96 Å It orders below the Curie temperature TC ≈ 16.5 K to a ferromagnetic state EuS consists of Eu2+ ions in the 8 S 7 ground state with a spin-only magnetic moment 2 of 7µB and therefore the spin-dependent part of the Coulomb interaction between two Eu2+ ions having localized spins is given by the Heisenberg Hamiltonian Inelastic neutron scattering investigations (Bohn . Rev. 181 92 0 ( 196 9). [2] O. Schärpf, Physica B 182 376 ( 199 2). [3] F. Tasset, P.J. Brown, E. Lelièvre-Berna, T. Roberts, S. Pujol, J. Alibon and E. Bourgeat-Lami, Physica B 267–268 69 ( 199 9). [4]. B.M. Wanklyn, J. Phys. C 21 291 7 ( 198 8). [8] D. Yablonski, Physica C 171 454 ( 199 0). [9] Yu.G. Raydugin, V.E. Naish and E.A. Turov, J. Magn. Magn. Mater. 102 331 ( 199 1). [10] P.J. Brown, T. Chattopadhyay,. 328 References 328 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji © 2006 Elsevier B.V. All rights reserved 245